MHB Find the Conjugate of a Denominator with Radicals

[bergausstein]

can you tell me what is the proper conjugate of the denominator.
what is the rule on how to group this kind of denominator to get the conjugate.$\displaystyle \frac{2+\sqrt{3}+\sqrt{5}}{2+\sqrt{3}-\sqrt{5}}$

thanks!

 

[mathbalarka]

I'd first multiply by the conjugate with respect to $\sqrt{5}$ and then multiply by the conjugate with respect to $\sqrt{3}$.

 

[bergausstein]

what do you mean by "with resperct to $\sqrt{5}$ and $\sqrt{3}$?

 

[mathbalarka]

First multiply the denominator by $2 + \sqrt{3} + \sqrt{5}$ to get $\left (2 + \sqrt{3}\right )^2 - 5 = 2 + 4\sqrt{3}$ and then multiply by $2 - 4\sqrt{3}$

 

[bergausstein]

my answer is

$\displaystyle \sqrt{3}+\frac{2\sqrt{5}}{11}+\frac{3\sqrt{15}}{22}$

is this correct?

 

[mathbalarka]

No. Can you check your calculations?

 

[SuperSonic4]

Don't forget you also need to multiply the numerator

$$\dfrac{2+\sqrt3+\sqrt5}{2+\sqrt3-\sqrt5}$$

$$\dfrac{2+\sqrt3+\sqrt5}{(2+\sqrt3)-\sqrt5} \times \dfrac{2+\sqrt3+\sqrt5}{(2+\sqrt3)+\sqrt5}$$

$$=\dfrac{(2+\sqrt3+\sqrt5)^2}{(2+\sqrt3)^2-5}$$

I would now expand $$(2+\sqrt3)^2$$ to get [math]7 + 4\sqrt(3)$$

If you put that in the fraction and collect like terms (ie: 7-5=2):

$$=\dfrac{(2+\sqrt3+\sqrt5)^2}{2+4\sqrt3}$$

Are you aware of the conjugate in this last expression? Generally when you rationalise the denominator nobody cares about the numerator - unless explicitly told otherwise leave it as is

 

[Deveno]

You can also do it "the other way":

$\dfrac{2+\sqrt{3}+\sqrt{5}}{2+\sqrt{3}-\sqrt{5}}$

$= \dfrac{2+\sqrt{3}+\sqrt{5}}{2+\sqrt{3}-\sqrt{5}}\cdot\dfrac{2-(\sqrt{3}-\sqrt{5})}{2-(\sqrt{3}-\sqrt{5})}$

$=\dfrac{6+4\sqrt{5}}{4 - (\sqrt{3} - \sqrt{5})^2}$

$=\dfrac{6+4\sqrt{5}}{4 - (8 - 2\sqrt{15})}$

$= \dfrac{6 + 4\sqrt{5}}{2\sqrt{15} - 4}$

At this point, you want to multiply by $2\sqrt{15} + 4$ top and bottom:

$= \dfrac{6 + 4\sqrt{5}}{2\sqrt{15} - 4}\cdot\dfrac{2\sqrt{15} + 4}{2\sqrt{15} + 4}$

whereupon you will arrive at an answer equivalent to SuperSonic4's (if you were to follow his next step).